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Let d(n) be the number of integers less then n which has an odd number of prime factors ( 2,3,5,7,8,11,12,13,17,18...).

How to prove d(n)/n have a limit 1/2?

Is there for all m an n such that $|n-2d(n)|>m$?

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For the second question, quite a bit more is known. There is a positive constant $k$ such that $|n-2d(n)|>k\sqrt{n}$ for infinitely many $n$. –  André Nicolas Apr 28 '12 at 19:29
    
what is the largest such k? –  user1708 Apr 28 '12 at 19:32

1 Answer 1

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You can begin to chase down things by starting at this link.

In particular, for the second question, quite a bit more is known. There is a positive constant $k$ such that $|n-2d(n)|>k\sqrt{n}$ for infinitely many $n$.

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